Noncommutative Algebras

9970413我们的研究旨在更好地理解某些有限生成的非交换代数的内部结构和表示理论。具体的激励例子包括李超代数的代数量子群和包络代数。本课程的重点主要是素数和本原理想理论，其基本目标是在此背景下将本原谱作为一种“非交换仿射代数几何”来研究。类似地，我们试图确定非对易Hopf代数的本原谱与仿射代数群之间的相似性；这一目标在很大程度上是由非对易Hopf代数在量子群理论中所起的作用所推动的。本文所涉及的技术问题包括态射、形变、链性、模扩张以及“小”的无限维量子群。这些抽象对象为研究非对易变量的多项式方程组提供了一个有效的环境；这样的方程是20世纪数学和物理的基础。我们在这项研究中的目的是研究一种由某些非对易方程组产生的“几何”，类似于从交换变量的多项式方程组的解得到的经典几何。

9970413Our research is directed toward better understanding the internal structure and representation theory of certain classes of finitely generated noncommutative algebras. The specific motivating examples include algebraic quantum groups and enveloping algebras of Lie superalgebras. The focus is mainly on prime and primitive ideal theory, and the underlying goal is to study primitive spectra in this context as a type of ``noncommutative affine algebraic geometry.'' Similarly, we seek to determine analogies between the primitive spectra of noncommutative Hopf algebras and affine algebraic groups; this latter objective is largely motivated by the role noncommutative Hopf algebras play in the theory of quantum groups. The technical issues we are concered with in this work include morphisms, deformations, catenarity, module extensions, and ``small'' infinite dimensional quantum groups.The subjects of our research are noncommutative associative algebras. These abstract objects have provided an effective setting for studying systems of polynomial equations in noncommuting variables; such equations are basic to twentieth century mathematics and physics. Our aim in this research is to study a type of ``geometry'' arising from certain noncommutative systems of equations, in analogy to the classical geometry obtained from solutions to systems of polynomial equations in commuting variables.